King at end of extra pilgrimage returned with eastern prince (5)
I believe the answer is:
rajah
'eastern prince' is the definition.
'king at end of extra pilgrimage returned' is the wordplay.
'king' becomes 'R' (abbreviation for rex, king in Latin).
'at end of' indicates one should take the final letters.
'pilgrimage' becomes 'haj' (haj is a kind of pilgrimage).
'returned' says the letters should be written in reverse.
The final letter of 'extra' is 'a'.
'haj' reversed gives 'jah'.
'r'+'a'+'jah'='RAJAH'
'with' is the link.
(Other definitions for rajah that I've seen before include "Royal Indian male" , "Indian king or prince" , "Hindu ruler" , "Former Indian ruler" , "Asian ruler" .)